6 days ago - Large circles should have smaller curvature than small circles ... of curvature, in this section we conside
Curvature (Com S 477/577 Notes)
Yan-Bin Jia Oct 3, 2017
We want to find a measure of how ‘curved’ a curve is. Since this ‘curvature’ should depend only on the ‘shape’ of the curve, it should not be changed when the curve is reparametrized. Further, the measure of curvature should agree with our intuition in simple special cases. Straight lines themselves have zero curvature. Large circles should have smaller curvature than small circles which bend more sharply. The (signed) curvature of a curve parametrized by its arc length is the rate of change of direction of the tangent vector. The absolute value of the curvature is a measure of how sharply the curve bends. Curves which bend slowly, which are almost straight lines, will have small absolute curvature. Curves which swing to the left have positive curvature and curves which swing to the right have negative curvature. The curvature of the direction of a road will affect the maximum speed at which vehicles can travel without skidding, and the curvature in the trajectory of an airplane will affect whether the pilot will suffer “blackout” as a result of the g-forces involved. In this lecture we will primarily look at the curvature of plane curves. The results will be extended to space curves in the next lecture.
1
Curvature
To introduce the definition of curvature, in this section we consider a unit-speed curve α(s), where s is the arc length. The tangential angle φ is measured counterclockwise from the x-axis to the unit tangent T = α′ (s), as shown below.
α(s)
φ(s)
x
The curvature κ of α is the rate of change in the direction of the tangent line at that point with respect to arc length, that is, dφ κ= . (1) ds 1
The absolute curvature of the curve at the point is the absolute value |κ|. Since α has unit speed, T · T = 1. Differentiating this equation yields T ′ · T = 0. The change of T (s) is orthogonal to the tangential direction, so it must be along the normal direction. The curvature is also defined to measure the turning of T (s) along the direction of the unit normal N (s) where T (s) × N (s) = 1. That is, T′ =
dT = κN. ds
(2)
We can easily derive one of the curvature definitions (1) and (2) from the other. For instance, if we start with (2), then κ = T′ · N dT = ·N ds T (s + ∆s) − T (s) = lim ·N ∆s→0 ∆s ∆φ · kT k = lim ∆s→0 ∆s ∆φ = lim ∆s→0 ∆s dφ . = ds
T (s + ∆s)
T (s) T (s) T (s + ∆s) N(s)
Example 1.
∆φ
Let us compute the curvature of the unit-speed circle s s . α(s) = r cos , sin r r
We obtain that
T
=
N
=
T′
=
s s , α′ (s) = − sin , cos r r s s − cos , − sin , r r s 1 s 1 N. α′′ (s) = − cos , sin = r r r r
2
Thus
1 . cf. (2) r The curvature of a circle equals the inverse of its radius everywhere. κ(s) =
The next result shows that a unit-speed plane curve is essentially determined once we know its curvature at each point of the curve. The meaning of ‘essentially’ here is ‘up to a rigid motion1 of R2 ’. Theorem 1 Let κ : (a, b) → R be an integrable function. Then there exists a unit-speed curve α : (a, b) → R2 whose curvature is κ. ˜ : (a, b) → R2 is any other unit-speed curve with the same curvature function κ, Furthermore, if α ˜ into α. there exists a rigid body motion that transforms α Proof
Fix s0 ∈ (a, b) and define, for any s ∈ (a, b), Z s φ(s) = κ(u) du, cf. (1), s0 Z s Z s α(s) = cos φ(t) dt, sin φ(t) dt . s0
s0
Then, the tangent vector of α is α′ (s) = cos φ(s), sin φ(s) ,
which is a unit vector making an angle φ(s) with the x-axis. Thus α is unit speed, and has curvature Z s d dφ = κ(u) du = κ(s). ds ds s0 For a proof of the second part, we refer to [3, p. 31]. The above theorem shows that we can find a plane curve with any given smooth function as its signed curvature. But simple curvature can lead to complicated curves, as shown in the next example. Example 2. we get
Let the signed curvature be κ(s) = s. Following the proof of Theorem 1, and taking s0 = 0,
φ(s)
=
α(s) =
s2 , 2 0 Z s Z s s2 s2 cos ds, sin ds . 2 2 0 0
Z
s
u du =
These integrals can only be evaluated numerically.2 The curve is drawn in the figure below.3 1
A rigid motion consists of a rotation and a translation. They arise in the theory of diffraction of light, where they are called Fresnel’s integrals, and the curve is called Cornu’s Spiral, although it was first considered by Euler. 3 Taken from [3, p. 33]. 2
3
When the curvature κ(s) > 0, the center of curvature lies along the direction of N (s) at distance from the point α(s). When κ(s) < 0, the center of curvature lies along the direction of −N (s) at distance − κ1 from α(s). In either case, the center of curvature is located at 1 κ
α(s) +
1 N (s). κ(s)
1 is called the radius of curvature. The osculating circle, when κ 6= 0, is the circle at the Here, |κ| 1 center of curvature with radius |κ| . It approximates the curve locally up to the second order.
center of curvature osculating circle radius of curvature 1/κ α(s)
The total curvature over a closed interval [a, b] measures the rotation of the unit tangent T (s) as s changes from a to b: Φ(a, b) =
Z
b
κ ds
a
=
Z
b
a
=
Z
dφ ds ds
b
dφ
a
= φ(b) − φ(a).
4
If the total curvature over [a, b] is within [0, 2π], it has a closed form: arccos T (a) · T (b) , if T (a) × T (b) ≥ 0; Φ(a, b) = 2π − arccos T (a) · T (b) , otherwise.
When the tangent makes several full revolutions4 as s increases from a to b, the total curvature cannot be determined just from T (a) and T (b). T (a) T (a)
α(a)
total curvature θ T (b) T (b) α(b)
A point s on the curve α is a simple inflection, or an inflection, if the curvature κ(s) = 0 but κ′ (s) 6= 0. Intuitively, a simple inflection is where the curve swing from the left of the tangent at the point to its right; or in the case of simple closed curve, it is where the closed curve α changes from convex to concave or from concave to convex. In the figure below, the curve on the left has one simple inflection while the curve on the right has six simple inflections. κ>0 κ0 κ>0
κ
